Principal-axis system orientation

The crystal structure of benzoic acid is monoclinic and contains four molecules per unit cell in the form of two magnetically inequivalent dimers with equal values of the chemical shift tensors but with their principal axis systems oriented... 

When using Eq. (11.11), care must be taken to rotate the constituent atomic susceptibility tensors, with their natural principal axis system oriented at the bond axes, into the principal inertia axis system of the molecule. With the components of the susceptibility tensor transforming like the corresponding coordinate products [compare Eq. (1.4)], the appropriate transformation is given by ... 

Because of the non-coincidence of the g- and -matrix principal axes, the various parallel features correspond to different orientations of the magnetic field in the g-matrix principal axis system. These orientations are given in Table 4.16. 

Fig. 10. Measurement of RDCs in a second, different alignment medium can significantly reduce the possible corresponding internuclear vector orientations. Permissible orientations will correspond to the intersection of cones, which are placed according to the respective principal axis systems of ordering.

The antisymmetric tensor is generally not observable in NMR experiments and is therefore ignored. The symmetric tensor is now diagonalized by a suitable coordinate transformation to orient into the principal axis system (PAS). After diagonalization there are still six independent parameters, the three principal components of the tensor and three Euler angles that specify the PAS in the molecular frame. 

As mentioned above, the principal values of chemical shift tensor give information about three dimensional electronic state of a molecule. However, in order to understand behavior of the principal values, one should obtain information about the orientation of the principal axis system of a chemical shift tensor with respect to the molecular fixed frame. The orientations of the principal axis systems of the chemical shift tensors of L-alanine Cp -carbons in some peptides were calculated, whose L-alanine moieties have different main-chain dihedral-angles, (( >,v /H-57.40,-47.50)[aR-helix], (-138.8°,134.7°)[ pA-sheet], (-66.3°,-... 

Figure 6. Model fragment used in 15N shielding calculation for imidazole. Also shown is the calculated orientation of the principal axis system of the shielding tensor. (Adapted from reference 26. Copyright 1991 American Chemical Society.)...

The calculated Euler angles (a = 50°, /3 = 60°, and y = 40°), which determine the relative orientation between the principal-axis system of the rotational diffusion tensor and that of the moment of inertia tensor, indicate a significant shift between the two tensors. This result is expected because of the fact that molecule 31 contains a number of polar groups and hydrogen-bonding centers, leading to strong intermolecular interactions. 

FIG. 12.—Orientation of the principal-axis system of inertia tensor (x. y. z1) and that of the rotational diffusion tensor (x,y,z) for compound 31. The principal diffusion axis x is perpendicular to the plane of the drawing. [Reproduced with permission from P. Dais, Carbohydrate Res., 263 (1994) 13-24, and Elsevier Science B.V.]... 

If I is oriented along the principal-axis system of the optical indicatrix, then for each of the three components of the refractive index (nlr n2, n3) the anisotropic Lorentz-Lorenz equation is... 

Figure 3 (A) The orientations of the principal axis system for EFG and CS tensors with respect to...

The frequency contribution from a CS tensor also depends on the molecular orientation with respect to the external magnetic field. As illustrated in Figure 3(A), the principal axis system for a CS tensor is defined with the direction of the magnetic field in this text. A liquid sample of H20 is generally used for chemical shift referencing (set to be 0 ppm). Similar to the case of an 170 EFG tensor, a quantum chemical approach is useful for the spectral analysis. Since quantum chemical calculations yield the CS, o, the following conversion is required for making a direct comparison between theoretical and experimental values ... 

The partitioned grand resistance matrix in Eq. (7.13) is a function only of the instantaneous geometrical configuration of the particulate phase. This consists of the fixed particle shapes together with the variable relative particle positions and orientations. As such, geometrical symmetry arguments (where such symmetry exists) may be used to reduce the number of independent, nonzero scalar components of the coefficient tensors in Eq. (7.13) for particular choices of coordinate axes (e.g., principal axis systems). 

Kottis and Lefebvre (322) have suggested that if polarized light is used to excite randomly oriented molecules to the triplet state, observation of the changes in the AMg = +1 ESR spectrum can reveal the correlation of the polarization properties of the excitation with the principal axis system of the triplet zero-field tensor. Such photoselection experiments have been carried out successfully by Lhotse and coworkers (323) and El-Sayed and Siegel (324) on a number of aromatic systems. Piette and collaborators (325) have studied the effect of metal complexation on the zero-field parameters and lifetimes of the phosphorescent triplet of aromatic-metal complexes with similar photoselection technique. The changes in... 

Finally, in a single crystal experiment it is possible to extract all five independent elements of the traceless NQI tensor, usnaUy chosen to be the orientation of the EFG principal axis system, in addition to the two parameters accessible for randomly oriented samples (jj and vq). This, of course, requires that it is possible to make a single crystal incorporating the radioactive mother nucleus, and is therefore limited to cases where the preparation can be completed within roughly the lifetime of the mother nucleus, for example, see Ref 138. 

Figure 1.7 Orientation of the magnetic field with respect to the g tensor principal axis system denoted X, Y, Z.

Figure 10 A snapshot of the solvation of xenon in acetonitrile together with an ellipsoidal representation of electric field gradient (EFG). The EFG ellipsoid is on average rhombic, and fluctuates both in form and orientation. The fluctuations in the eigenvalues gives a fast vibrational averaging, and the long time decay in determined by reorientation of the EFG principal axis system.

The orientational dependence is present in the chemical shielding tensor which in its principal axis system (PAS) is diagonal with principal components axx, o yy and ctzz-In a crystalline powder sample all orientations of the PAS system of the shielding tensor are present and for most orientations the PAS does not coincide with the laboratory (LAB) frame where the z-axis is aligned with the direction of the applied magnetic field Bq. Hence the tensor representation gives in the PAS... 

Figure 2.10. The relative orientations of the principal axis system (PAS), rotor (ROT) and...

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